1 Seepage as a model of counter-terrorism - PowerPoint Presentation

Slide1

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Seepage as a model of counter-terrorism

Anthony BonatoRyerson University

CMS Winter Meeting

December 2011Slide2

Good guys vs bad guys games in graphs

2

slow

medium

fast

helicopter

slow

traps, tandem-winmediumrobot vacuumCops and Robbersedge searchingeternal securityfastcleaningdistance k Cops and RobbersCops and Robbers on disjoint edge setsThe Angel and DevilhelicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil,FirefighterHex

bad

goodSlide3
Seepage

Seepage

3motivated by the

1973 eruption of the Eldfell volcano in Icelandto protect the harbour, the inhabitants poured water on the lava in order to solidify and halt itSlide4

Seepage

(Clarke,Finbow,Fitzpatrick,Messinger,Nowakowski,2009)

greens and sludge, played on a directed acylic graph (DAG) with one source s

the players take turns, with the

sludge

going first by contaminating

s

on subsequent moves sludge contaminates a non-protected vertex that is adjacent to a contaminated vertexthe greens, on their turn, choose some non-protected, non-contaminated vertex to protectonce protected or contaminated, a vertex stays in that state to the end of the gamesludge wins if some sink is contaminated; otherwise, the greens winSeepage4Slide5
Example 1:

G1

Seepage

5S

G

G

S

G

GSSlide6

Example 2:

G2Seepage

6S

G

G

S

xSlide7
Green number

green number

of a DAG G, gr(G), is the minimum number of greens needed to win

gr(G) = 1: G is green-winprevious examples: gr(G1

) = 3,

gr

(G

2

) = 1(CFFMN,2009): characterized green-win treesbounds given on green number of truncated Cartesian products of pathsSeepage7Slide8
Characterizing trees

in a rooted tree

T with vertex x, Tx is the

subtree rooted at xa rooted tree T is green-reduced to T − Tx

if

x

has out-degree at 1 and every ancestor of

x

has out-degree greater than 1T − Tx is a green reduction of TTheorem (CFFMN,2009)A rooted tree T is green-win if and only if T can be reduced to one vertex by a sequence of green-reductions. Seepage8Slide9
Mathematical counter-terrorism

(Farley et al. 2003-)

: ordered sets as simplified models of terrorist networksthe maximal elements

of the poset are the leaderssubmit plans down via the edges to the foot soldiers or minimal nodes only one messenger needs to receive the message for the plan to be executed.considered finding minimum order cuts: neutralize operatives in the network

Seepage

9Slide10
Seepage as a counter-terrorism model?

seepage has a similar paradigm to model of

(Farley, et al)main difference: seepage is dynamicas messages move down the network towards foot soldiers, operatives are neutralized over time

Seepage10Slide11
Structure of terrorist networks

competing views;

for eg (Xu et al, 06), (Memon, Hicks, Larsen, 07), (Medina,Hepner,08)

:complex network: power law degree distributionsome members more influential and have high out-degreeregular network:

members have constant out-degree

members are all about equally influential

Seepage

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Our model

we consider a stochastic DAG model

total expected degrees of vertices are specifieddirected analogue of the G(w) model of Chung and Lu

Seepage12Slide13

Seepage13

let w = (w

1

, …,

w

n

) be a sequence G(w): probability space of graphs on [n], where i and j are joined independently with probability G(w) is the space of random graphs with given expected degree sequence w if w = (pn,…,pn), then G(w) is just G(n,p) if w follows a power law: random power law graphsRandom graphs with given expected degree sequence (Chung, Lu, 2003)Slide14
General setting for the model

given a DAG

G with levels Lj, source v, c > 0

game G(G,v,j,c): nodes in Lj

are sinks

sequence of discrete time-steps

t

nodes protected at time-step tgrj(G,v) = inf{c ϵ N: greens win G(G,v,j,c)}Seepage14Slide15

Random DAG model (Bonato, Mitsche, Prałat,11+)

parameters: sequence (w

i : i > 0), integer nL

0

= {v};

assume

L

j definedS: set of n new verticesdirected edges point from Lj to Lj+1 a subset of Seach vi in Lj generates max{wi -deg-(vi),0} randomly chosen edges to Sedges generated independentlynodes of S chosen at least once form Lj+1 parallel edges possible (though rare in sparse case)Seepage15Slide16

d-regular case

for all i, wi = d > 2

a constantcall these random d-regular DAGsin this case, |Lj| ≤ d(d-1)

j-1

we give bounds on

gr

j(G,v) as a function of the levels j of the sinksSeepage16Slide17
Main results

Theorem

(BMP,11+) :If G is a random d-regular DAG, then a.a.s. the following hold.

If 2 ≤ j ≤ O(1), then grj

(G,v) = d-2+1/j.

If

ω

is any function tending to infinity with

n and ω ≤ j ≤ logd-1n- ωloglog n, then grj(G,v) ≤ d-2.If logd-1n- ωloglog n ≤ j ≤ logd-1n - 5/2klog2log n + logd-1log n-O(1) for some integer k>0, then d-2-1/k ≤ grj(G,v) ≤ d-2.Seepage17Slide18

grj

(G,v) is smaller for larger jTheorem (BMP,11+) For a random d

-regular DAG G, for s ≥ 4 there is a constant

C

s

> 0

, such that if

j ≥ logd-1n + Cs,then a.a.s. grj(G,v) ≤ d - 2 - 1/s.proof uses a combinatorial-game theory type argumentSeepage18Slide19
Sketch of proof

greens protect

d-2 vertices on some layers; other layers (every si steps, for

i ≥ 0) they protect d-3greens play greedily: protect vertices adjacent to the sludge≤1 choice for sludge when the greens protect d-2; at most 2,

otherwise

greens can move sludge to any vertex in the

d-2

layers

bad vertex: in-degree at least 2if there is a bad vertex in the d-2 layers, greens can directs sludge there and sludge losesgreens protect all childrenSeepage19t = si+1d-3Slide20
Sketch of proof, continued

sludge wins implies that there are no bad vertices in

d-2 layers, and all vertices in the d-3 layers either have in-degree 1

and all but at most one child are sludge-win, or in-degree 2 and all children are sludge-winallows for a cut

proceeding inductively from the source to a sink:

in a given

d-3

layer, if a vertex has

in-degree 1, then we cut away any out-neighbour and all vertices not reachable from the source (after the out-neighbour is removed)if sludge wins, then there is cut which gives a (d-1,d-2)-regular graphthe probability that there is such a cut is o(1)Seepage20d-3Slide21
Power law case

fix

d, exponent β > 2, and maximum degree M = n

α for some α in (0,1)wi = ci

-1/

β

-1

for suitable c and range of ipower law sequence with average degree dideas:high degree nodes closer to source, decreasing degree from left to rightgreens prevent sludge from moving to the highest degree nodes at each time-stepSeepage21Slide22

Theorem (BMP,11+)In a random power law DAG:

Seepage

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Contrasting the cases

hard to compare

d-regular and power law random DAGs, as the number of vertices and average degree are difficult to controlconsider the first case when there is Cn vertices in the d

-regular and power law random DAGsmany high degree vertices in power law casegreen number higher than in d-regular caseinterpretation: in random power law DAGs, more difficult to disrupt the network

Seepage

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Open problems

in

d-regular case, green number for j between logd-1

n - 5/2log2log n + logd-1log n-O(1) and logd-1

n + c

?

other sequences?

infinite case:

grj(G,v) is non-increasing with j and bounded, so has a limit g(G,v)seepage on:infinite acyclic random oriented graph (Diestel et al, 07)infinite semi-directed graphs with constant out-degree (B, Delic, Wang,11+)Seepage24Slide25

Seepage

25preprints, reprints, contact:

search: “Anthony Bonato”Slide26

Seepage

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